Piezoresistance effect in n-type silicon: from bulk to nanowires

Abstract

The first order piezoresistance coefficients are examined in the n-type silicon structures with different dimensionality of electron gas: bulk crystal, quantum film (well) and quantum wire. The detail research involves quantum kinetic approach to calculation of the kinetic coefficients (conductivity, mobility, concentration) of electrons in the strained and unstrained states. As scattering system were adopted ionized impurities, longitudinal acoustic phonons and surface roughness. Detailed studies have been carried out for dependences of electron mobility and piezoresistance coefficients on confining dimensions. An alternative explanation is proposed for origin of the giant piezoresistance effect in n-type silicon nanostructures. Comparison of the obtained results shows not only qualitative but even sufficient quantitative agreement with experimental data.

Appendices

Appendix 1

Expression (10) for kinetic tensor includes the spectral correliators of electron collision with external system of the scattering centers $\langle \varphi ^{2}\rangle _{\omega , \vec {q}}^{(D)}$. Below we will consider scattering by longitudinal acoustic phonons, ionized impurities for bulk material ($D = 3$) and for low-dimensional nanostructures ($D = 1, 2$) we will take into consideration the surface roughness scattering. We will assume that in quantum film ($D = 2$) and quantum wire ($D = 1$) the electron motions are quantized accordingly in $z$-direction and in z-, y- directions. In the first case electrons move free in xy-plane and in the second along $x$-direction only.

1.1 Scattering by longitudinal acoustic phonons

1.1.1 Bulk material

For longitudinal acoustic phonons with dispersion law $\omega = s_{L} q (s_{L}$ is the longitudinal sound velocity) the spectral correlator is [5, 8, 9]

$$\begin{aligned} \langle \varphi ^{2}\rangle _{Ac}^{(D=3)}{}_{\omega ,\vec {q}}^{(v)}&= \left( {\frac{\Xi _v}{e}}\right) ^{2}\frac{\pi \hbar \omega }{2\rho s_L^{2}}\coth \left( {\frac{\hbar \omega }{2k_B T}}\right) \left[ \delta (\omega -s_L q)\right. \nonumber \\&\left. +\,\,\delta (\omega +s_L q)\right] , \end{aligned}$$

(57)

$\Xi _{v}$ is the acoustic deformation potential, $\rho$ is the density of crystal, the $\delta$-function ensures the energy conservation at emission and absorption of phonons. For bulk material we take into account the angular dependence of the deformation potential [9, 18, 37] and expression for deformation potential could be represented as $\Xi _{v}^{2} = \Xi _{d}^{2} Q^{(v)}_{xx(yy,zz)}$. The values $Q^{(v)}_{xx(yy,zz)}$ for the different valleys are given in Table 1, where $Q_{\parallel }(L)$ and $Q_{\bot }(L)$ [9] are

$$\begin{aligned}&Q_\parallel (L)=1+\frac{\Xi _u }{\Xi _d }\frac{2L}{(L-1)^{2}}\left[ {L-4+\frac{3}{\sqrt{L-1}}\arctan (\sqrt{L-1})}\right] \nonumber \\&\quad +\,\frac{3}{2}\left( {\frac{\Xi _u }{\Xi _d }}\right) ^{2}\frac{L}{(L-1)^{3}} \left[ \frac{2L^{2}-14L-3}{3}+\frac{5L\arctan (\sqrt{L-1})}{\sqrt{L-1}}\right] .\nonumber \\ \end{aligned}$$

(58)

$$\begin{aligned}&Q_\bot (L)=1+\frac{\Xi _u }{\Xi _d }\frac{9}{4(L-1)^{2}}\left[ {\frac{2-5L}{3}+\frac{L^{2}\arctan (\sqrt{L-1})}{\sqrt{L-1}}}\right] \nonumber \\&\qquad \qquad \qquad +\,\,\frac{15}{32}\left( {\frac{\Xi _u }{\Xi _d}}\right) ^{2}\frac{1}{(L-1)^{3}}\left[ L^{2}+\frac{16 L}{3}-\frac{4}{3}\right. \nonumber \\&\qquad \qquad \qquad \left. +\,\,L^{2}(L-6)\frac{\arctan (\sqrt{L-1})}{\sqrt{L-1}}\right] . \end{aligned}$$

(59)

Here $L = m_{\parallel }/m_{\bot }$.

As it is shown [5–8, 38] the spectral correlators in low-dimensional semiconductor structures could be obtained by integration

$$\begin{aligned}&\left\langle {\varphi ^{2}} \right\rangle ^{(D=2)}_{\omega , q_\bot } =\frac{1}{2\pi }\int \limits _{-\infty }^\infty {\left\langle {\varphi ^{2}} \right\rangle }_{\omega ,\vec {q}}^{(D=3)} dq_z , \quad q_{\bot }\in \{q_{x}, q_{y}\}, \end{aligned}$$

(60)

$$\begin{aligned}&\left\langle {\varphi ^{2}} \right\rangle ^{(D=1)}_{\omega , q_x} =\frac{1}{2\pi }\int \limits _{-\infty }^\infty {\left\langle {\varphi ^{2}}\right\rangle }_{\omega , q_\bot }^{(D=2)} dq_y. \end{aligned}$$

(61)

After integration we obtain correliators in quantum films and quantum wires in the following forms [5–8, 38]

$$\begin{aligned}&\langle \varphi _{Ac}^{2}\rangle ^{(D=2)}_{\omega ,q_\bot } =\left( {\frac{\Xi _A}{e}}\right) ^{2}\frac{\hbar \omega ^{2}}{2\rho s^{3}}\coth \left( {\frac{\hbar \left| \omega \right| }{2k_B T}}\right) \frac{\Phi \left( {\omega ^{2}-q_\bot ^{2}s^{2}}\right) }{\sqrt{\omega ^{2}-q_\bot ^{2}s^{2}}},\nonumber \\ \end{aligned}$$

(62)

$$\begin{aligned}&\langle \varphi _{Ac}^{2}\rangle ^{(D=1)}_{\omega ,q_x } =\left( {\frac{\Xi _A}{e}}\right) ^{2}\frac{\hbar \omega ^{2}}{4\rho s^{4}}\coth \left( {\frac{\hbar \left| \omega \right| }{2k_B T}}\right) \Phi \left( {\omega ^{2}-q_x^{2}s^{2}}\right) .\nonumber \\ \end{aligned}$$

(63)

$\Phi (x)$ is the Heaviside step function.

In the low dimensional silicon structures we will use the averaged value of the acoustic deformation potential $\Xi _{A}^{2}=\Xi _{d}^{2}+\Xi _{d} \Xi _{u}+3\Xi _{u}^{2}/8$ [9].

1.2 Scattering by ionized impurities

Coulomb potentials of the point unitary charged impurity in bulk material, quantum film and quantum wire could be written as [5, 7, 39–41]

$$\begin{aligned}&\varphi ^{(D=3)}=\frac{4\pi e}{\varepsilon _L \left| {\vec {q}} \right| ^{2}}, \end{aligned}$$

(64)

$$\begin{aligned}&\varphi ^{(D=2)}=\frac{1}{2\pi }\int \limits _{-\infty }^\infty {\varphi ^{(D=3)}} dq_z =\frac{2\pi e}{\varepsilon _L q_\bot }, \quad q_{\bot } \in \{q_{x}, q_{y}\},\nonumber \\ \end{aligned}$$

(65)

$$\begin{aligned}&\varphi ^{(D=1)}=\frac{1}{2\pi }\int \limits _{-\infty }^\infty {\exp \left( {-\frac{q_y^{2}W_y^{2}}{2}}\right) \varphi ^{(D=2)}} dq_y, \end{aligned}$$

(66)

$\varepsilon _{L}$ is the dielectric constant of the lattice. In (67) we have to introduce the regularization factor exp ($-q_{y}^{2}W_{y}^{2}/2$)(see, more [5, 7, 10, 39, 40]). In this case a quantum wire with a finite but small extension $W_{y}$ in the quantum-confined $y$-direction is considered. After integration (67) we have

$$\begin{aligned} \varphi ^{(D=1)}=\frac{e}{\varepsilon _L }\exp \left( {\frac{q_x^{2}W_y^{2}}{4}}\right) K_0 \left( {\frac{q_x^{2}W_y^{2}}{4}}\right) , \end{aligned}$$

(67)

$K_{0}$ is the zeroth-order modified Bessel function. At small transferred momentum $q_{x}^{2}W_{y}^{2}/4\le 1$ expression (68) could be simplified [5, 39]

$$\begin{aligned} \varphi ^{(D=1)}=2\frac{e}{\varepsilon _L }\ln \left( {\frac{q_1 \sqrt{8}}{\left| {q_x}\right| }}\right) . \end{aligned}$$

(68)

Correlators of scattering potentials we obtain by averaging over positions of the scattering centers and performing Fourier transform in time [5, 7, 9]. For bulk material we take into account contribution of electrons $\Delta \varepsilon ^{(0)}(\omega = 0, q) = \varepsilon _{L} q_{0}^{2}/q^{2}$ in the screening process [5, 8, 9, 38]

$$\begin{aligned} \langle \varphi ^{2}\rangle ^{(D=3)}_{\omega ,\vec {q}} =\frac{2^{5}\pi ^{3}N_I^{(D=3)} e^{2}}{\left[ \varepsilon _L +\Delta \varepsilon ^{(0)}(\omega =0,q)\right] ^{2}q^{4}}\delta \left( \omega \right) , \end{aligned}$$

(69)

where $\Delta \varepsilon ^{(0)}(0,q)=\varepsilon _L \frac{q_0^2}{q^{2}}, q_0^2 =\frac{12e^{2}m_\bot }{\hbar ^{3}\varepsilon _L} \left[ {\frac{2m_{||} k_B T }{\pi }}\right] ^{1 / 2}$ $F_{- 1/2} (\eta _v \left( {\hat{{e}}}\right) )$.

In contrast to bulk material there is no exponential screening in low-dimensional semiconductor structures and we will neglect by it (see, more [5, 6, 39, 41, 42]).

$$\begin{aligned}&\left\langle \varphi ^{2}\right\rangle _{\omega ,q_\bot }^{^{(D=2)}} =\frac{\left( {2\pi }\right) ^{3}e^{2}N_I^{(D=2)}}{\varepsilon _L^{2}q_\bot ^{2}}\delta \left( \omega \right) , \end{aligned}$$

(70)

$$\begin{aligned}&\left\langle {\varphi ^{2}} \right\rangle _{\omega .q_x }^{(D=1)} =\frac{8\pi ^{3}e^{2}N_I^{(D=1)}}{\varepsilon _L^{2}}\left[ {\ln \left( {\frac{\sqrt{8}}{W_y \left| {q_x } \right| }}\right) }\right] ^{2}\delta \left( \omega \right) . \end{aligned}$$

(71)

$N_{I}^{(D=2)}$ and $N_{I}^{(D=1)}$ are accordingly the areal and linear densities of scattering centers. For comparison with experiment in which nanostructures with different thickness but fixed bulk concentration of ionized impurities are investigated (see, for instance [2]) we will assume $N_{I}^{(D=3)} = N_{I}^{(D=2)}/W_{z}=N_{I}^{(D=1)}/W_{z}W_{y}$.

1.3 Surface roughness scattering

1.3.1 Quantum film

The interface roughness scattering potential $\varphi \left( {\vec {r}_\bot }\right) ^{(v)}$ is given by a fluctuation of the quantized energy of electrons in the valley $v$ with width fluctuation $\Delta ({\vec {r}})$ of the quantum film thickness [10, 13, 43]

$$\begin{aligned} \varphi \left( {\vec {r}_\bot }\right) ^{(v)}=\frac{1}{e}\left| {\frac{\partial \varepsilon _{\vec {k}}^{(v)}}{\partial W_z }} \right| \Delta _f \left( {\vec {r}_\bot }\right) . \end{aligned}$$

(72)

The surface roughness is assumed to be characterized by the lateral size $\Lambda _{f}$ and height $\Delta _{f}$ of Gaussian fluctuations of the thickness [10, 13, 43, 44]. In this case the space-time correlator is

$$\begin{aligned} K_f ({\vec {r}_\bot , t})=\left\langle {\Delta _f \left( {\vec {r}_\bot }\right) \Delta _f \left( {{\vec {r}}'_\bot }\right) } \right\rangle =\Delta ^{2}\exp \left( {-\frac{\vec {r}_\bot ^{2}}{\Lambda _f^{2}}}\right) . \end{aligned}$$

(73)

The spectral correlator of the scattering potential could be obtained by Fourier transform [45]

$$\begin{aligned}&\left( {\left\langle {\varphi ^{2}} \right\rangle _{\omega , \vec {q}_\bot }^{(v)}}\right) _{SR}^{(D=2)} =\left[ {\frac{1}{e}\frac{\partial \varepsilon _{\vec {k}}^{(v)}}{\partial W_z }}\right] ^{2}\int \limits _{-\infty }^\infty {\exp \left( {-i\omega t}\right) dt}\nonumber \\&\quad \times \int \limits _{-\infty }^\infty {K_f \left( {\vec {r}_\bot ,t}\right) \exp \left( iq_\bot \vec {r}_\bot \right) } d^{2}\vec {r}_\bot . \end{aligned}$$

(74)

Finally we have

$$\begin{aligned} \left\langle {\varphi ^{2}} \right\rangle _{\omega ,\vec {q}_\bot }^{(v)}=2\pi ^{2}\left[ {\frac{\partial \varepsilon _{\vec {k}}^{(v)}}{e\partial W_z}}\right] ^{2}\Delta _f^{2}\Lambda _f^{2}\exp \left( {-\frac{q_\bot ^2 \Lambda _f^{2}}{4}}\right) \delta \left( \omega \right) .\nonumber \\ \end{aligned}$$

(75)

1.3.2 Quantum wire

The space-time correlator for quantum wire is [45–49]

$$\begin{aligned} K_w ({r_x ,t})=\left\langle {\Delta _w (r_x)\Delta _w \left( {r_x^{\prime }-r_x }\right) } \right\rangle =\Delta _{w}^2 \exp \left( {-\frac{\sqrt{2}\left| {r_x } \right| }{\Lambda _w}}\right) ,\nonumber \\ \end{aligned}$$

(76)

$\Delta _{w}$ is the root-mean square fluctuation of the roughness, $\Lambda _{w}$ is the correlation length.

Performing the Fourier transforms with respect to time and $r_{x}$ we obtain

$$\begin{aligned}&\left( {\left\langle {\varphi ^{2}} \right\rangle _{\omega , \vec {q}_\bot }^{(v)}}\right) _{SR}^{(D=1)} =\left[ {\left( {\frac{\partial \varepsilon _{\vec {k}}^{(v)}}{e\partial W_z }}\right) ^{2}+\left( {\frac{\partial \varepsilon _{\vec {k}}^{(v)}}{e\partial W_y}}\right) ^{2}}\right] \nonumber \\&\quad \times \,\,\frac{4\sqrt{2}\pi \Delta _w^2 \Lambda _w}{\left( {2+q_x^{2}\Lambda _w^{2}}\right) }\delta (\omega ). \end{aligned}$$

(77)

As it follows from dispersion relations (15)–(17)

$$\begin{aligned} \left[ {\frac{1}{e}\frac{\partial \varepsilon _{\vec {k}}^{(v)}}{\partial W_{z(y)}}}\right] ^{2}=\left( {\frac{\left( {\hbar \pi l_{z(y)} }\right) ^{2}}{em_{zz(yy)}^{(v)}W_{z(y)}^{3}}}\right) ^{2}. \end{aligned}$$

(78)

In calculations we will assume $\Delta _{f} = 1.03\,\mathrm{nm}, \Lambda _{f} = 0.27$ nm [45] and $\Delta _{w} = 0.48\,\mathrm{nm},\; \Lambda _{w} = 1.3\,\mathrm{nm}$ [49].

Appendix 2

In this section we evaluate integral $I_{v}(q)^{(D)}$ for electron gas with different dimensionality $D \in 1, 2, 3$

$$\begin{aligned} I_v (q)^{(D)}=\int {\delta \left( {\varepsilon _{\vec {k}}^{(v)}-\varepsilon _{\vec {k}-\vec {q}}^{(v)}-\hbar \omega }\right) \frac{\partial f_{\vec {k}}^0 }{\partial \varepsilon _{\vec {k}} }^{(v)}d^{D}\vec {k}}. \end{aligned}$$

(79)

Substituting the relevant dispersion relations from (15)–(17) into integral (80) we derive

$$\begin{aligned}&I_v \left( {q_x ,q_y ,q_z ,\hat{{e}}}\right) ^{(D=3)}\nonumber \\&\quad =-\frac{2\pi \sqrt{m_{xx}^{(v)} m_{yy}^{(v)} m_{zz}^{(v)}}}{\hbar ^{4}}\left( {\frac{q_x^2 }{m_{xx}^{(v)} }+\frac{q_y^2 }{m_{yy}^{(v)}}+\frac{q_z^2 }{m_{zz}^{(v)} }}\right) ^{-1/2}\nonumber \\&\qquad \times \left\{ 1+\exp \left[ \frac{\hbar ^{2}}{8k_B T}\left( {\frac{q_x^2 }{m_{xx}^{(v)} }+\frac{q_y^2 }{m_{yy}^{(v)} }+\frac{q_z^2 }{m_{zz}^{(v)} }}\right) \right. \right. \nonumber \\&\qquad \quad \left. \left. -\,\eta _v \left( {\hat{{e}}}\right) ^{(D=3)}\right] \right\} ^{-1}. \end{aligned}$$

(80)

$$\begin{aligned}&I_v \left( {q_x ,q_y ,\hat{{e}}}\right) ^{(D=2)}\nonumber \\&\quad =-\sqrt{\frac{2}{\pi }}\frac{m_{xx}^{(v)}m_{yy}^{(v)}}{\hbar ^{3}\sqrt{m_{yy}^{(v)} q_x^{2}+m_{xx}^{(v)}q_y^{2}}\sqrt{k_B T}} F_{-3/2} \left( {\eta _v^{(D=2)}\left( {\hat{{e}},l_z }\right) }\right) \nonumber \\ \end{aligned}$$

(81)

$$\begin{aligned}&I_v \left( {q_x ,\hat{{e}},l_z ,l_y}\right) ^{(D=1)}\nonumber \\&\quad =-\frac{m_{xx}^{(v)}}{\left| {q_x } \right| \hbar ^{2}k_B T}\exp \left( {\frac{q_x^{2}\hbar ^{2}}{8m_{xx}^{(v)}k_B T}-\eta _v^{(D=1)}\left( {\hat{{e}},l_z ,l_y }\right) }\right) \nonumber \\&\qquad \times \left[ {\exp \left( {\frac{q_x^{2}\hbar ^{2}}{8m_{xx}^{(v)}k_B T}-\eta _v^{(D=1)}\left( {\hat{{e}},l_z ,l_y }\right) }\right) +1}\right] ^{-2}. \end{aligned}$$

(82)

Here $\eta _v \left( {\hat{{e}}}\right) ^{(D=3)}=\frac{\varepsilon _F -\Delta \varepsilon _v \left( {\hat{{e}}}\right) }{k_B T}$, $\eta _v^{(D=2)}\left( {\hat{{e}},l_z }\right) =\frac{\varepsilon _F -\Delta \varepsilon _v \left( {\hat{{e}}}\right) }{k_B T}$ $-\frac{\hbar ^{2}}{2m_{zz}^{(v)}k_B T}\left( {\frac{\pi l_z }{W_z }}\right) ^{2}, \eta _v^{(D=1)}\left( {\hat{{e}},l_z ,l_y }\right) =\frac{\varepsilon _F -\Delta \varepsilon _v \left( {\hat{{e}}}\right) }{k_B T}-\frac{\hbar ^{2}}{2k_B Tm_{yy}^{(v)}}$ $\left( {\frac{\pi l_y }{W_y }}\right) ^{2}-\frac{\hbar ^{2}}{2k_B Tm_{zz}^{(v)}}\left( {\frac{\pi l_z}{W_z}}\right) ^{2}$ and $l_{z}, l_{y} = 1, 2, \ldots$